Is this likely to happen?
Until the 1990s, the most dominant language of the EU (European Union) was French. When the EU was the EC (European Community) and the official language policy was defined, Dutch, French, German and Italian were identified as the working languages. However, as more countries joined, many of which had English as a second or additional language, the number of English speakers grew until English was the majority common language.
Image from jakubmarian.com
Currently, the EU lists 24 official and working languages. The UK is the only member country which gives English as its official language, but English is the most commonly used language in EU debates and discussions. There are a few member countries which commonly use English but have nominated a different language as their EU official language; for example, the Republic of Ireland gives Irish Gaelic as its official language, and Malta gives Maltese.
If Britain withdraws from the EU, there will be no member country listing it as an official language. (There is of course the possibility that England will withdraw and e.g. Scotland and Northern Ireland will remain; it will be interesting to see what happens linguistically in that instance.) In order for English to continue to be used as an official language, all remaining members of the EU will have to agree*.
But does choice of language work like that?
Historically, English has weathered a number of storms. When members of the British Empire sought to gain their independence, it may have seemed logical for English - the language of the colonial oppressors - to be rejected at the same time. The fact that this did not happen and that English is used as a first or second language in more than 70 countries worldwide points to the usefulness of English as a global language, but also to its developing socio-economic and political status during the 20th century. With the decline of the British Empire came the rise of the United States of America, which has English as its official language. One of my colleagues, Dr Lynne Murphy from the University of Sussex, regularly presents on how America saved the English language; from the perspective of its use as a global lingua franca, she's got a point. (If you're interested, you can follow Lynne's wonderful blog, Separated by a Common Language, and find her on Twitter @Lynneguist.)
In fact, in some post-colonial situations, English is regarded as a more or less neutral language. In India, for example - and this is an oversimplified summary - English was to be phased out over a period of 50 years post-independence in 1947 in favour of Hindi. However, as not everyone in India speaks Hindi, and many do not want to for various cultural reasons, English continued to be used, and is now an official language of India. In Hong Kong, there was quite a strong desire at the time of the Handover in 1997 for the British to stay and for the territory not to be handed back to China. The fact that English is still an official language of Hong Kong may reflect this desire, but also it has its uses in Hong Kong, which is an international hub for trade and finance. Singapore has Malay, Chinese (various dialects) and Tamil speakers, among others; English is a unifying language.
But these Englishes we are talking about here are not 'British English', or even 'American English'. The Englishes spoken around the world may be based on one or other variety, but they have developed their own vocabulary and grammar. Euro-English is no exception; there is research into this variety, and even guides on how to use EU English. English simply does not belong to traditional 'native' English speakers any more; it belongs to everyone who speaks it, and communities will enact development to fit need and use. Brits and Americans need to bear this in mind when using English to interact in international settings, as they cannot assume they will be understood by every English speaker.
So, will English cease to be a language of the EU? Probably not, either in conversations between EU member countries, MEPs, or in EU interaction with other countries around the world**. It is simplistic to think it can simply be voted out. After all, English is much more than just the language of the United Kingdom.
* I had understood that there would have to be a vote to keep English as a language of the EU. The opposite is, in fact, true: there would have to be a unanimous vote to remove it as an official language, as clarified in this statement on behalf of the European Commission in Ireland, dated 28th June 2016.
** One MEP from Sweden suggests that communications in the EU could be fairer in English, as it will be everyone's second language.